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Sāmānya (सामान्य), Viśeṣa (विशेष), and Samavāya (समवाय): The Mathematical Expressions

Sāmānya (Set):

The English word for the word ‘Sāmānya’ of Vaiśeṣika (वैशेषिक) is ‘set’. Vaiśeṣikaśāstra has defined the property of Sāmānya in the following Sūtra:

अनुवृत्तिप्रत्ययकारणत्वं सामान्यत्वम् ||

It means that the cause of assimilative cognition is called Sāmānya or the set. According to Vaiśeṣikaśāstra one set comprises the physical entities which are similar or homogenous in shape or kind.

आकारप्रकारैक्यसाधर्म्यादिबोधकारणीभूता या जातिः सा सामान्यपदेन व्यपदिश्यते | व्याप्यत्वव्यापकत्वादिना सामान्यस्य वैशेषिक दर्शने त्रयो भेदाः समाम्नाताः सन्तिः

१. परम् २. अपरम् ३. परमपरम् चेति।

This aphorism is very useful for elaborating and discussing the concept of the set. In mathematics, numbers may be considered to form a set. The sets can be used for developing subsets and classes (elements) by using simple mathematical operations like addition and subtraction. Using algebraic quantities, sets of different kinds may be formed. By assigning physical meaning to these algebraic quantities and correlating them with numbers, one can develop algebra. We know that the addition or subtraction of quantities of same species or physical quantities is logically possible. Let us for example assume that at one place there are 4 human beings, 15 cows and 6 buffaloes, (in all 25 living beings), and 21 of them are animals. In this example, we may classify this set into a universal set (parasāmānya, परसामान्य) – comprising all 25 living beings, one subset (parāparasāmānya, परापरसामान्य) comprising 21 animals: cows and buffaloes, and another subset comprising 4 human being. The human beings, cows and buffaloes are known as the elements (aparasāmānya, अपरसामान्य) of the
set.
Similarly, if at another place, there are 2 human beings, 3 cows and 5 buffaloes, by applying the addition operator on these two sets, the numbers of living beings, human beings and animals can be obtained by adding the qualifying numbers of universal set, elements and subset. If an element is not there in a set, or the elements are removed from the first set, the difference needs to be considered.

This can be algebraically put as follows:
If we have L for living being, H for human being, C for Cow, B for Buffalo and A for animal, then

The first set = 4 H + 15 C + 6 B
The second set = 2 H + 3 C + 5 B
Their sum = 6 H + 18 C + 11 B
The difference = 2 H + 12 C + B
If the count of living beings is required, L being the universal set, algebraically we have
H = L, C = L and B = L
Therefore, on substitution we can get the total count of living beings
= 6 L + 18 L + 11 L = 35 L = 35 living beings.
Similarly, the total number of animals (animal forming a subset) can be obtained if we substitute C = A and B = A, then the total number of animals = 18 A + 11 A = 29 animals and so on.

Viśeṣa (Specific):

Vaiśeṣikaśāstra has called the physical quantity as Viśeṣa in

अत्यन्तव्यावृत्तिबुद्धिहेतुर्विशेष इति |

The cause of cognition of ultimate difference is called Viśeṣa (specific). The physical entity bearing no parts, which is of eternal character and is present for ever, is termed as ‘Viśeṣa dravya (विशेष द्रव्य)’. It is important here to note that such partless entities are central to the concept of Vaiśeṣika philosophy, as it is clear from the name ‘Vaiśeṣika’. The physical quantity bearing ultimate and partless character is termed as Viśeṣa dravya (specific physical quantities or the fundamental physical quantities). According to Vaiśeṣika, such partless quantities are plasma (ākāśa, आकाश), length vector or space (dik, दिक्), time (kāla, काल), soul (ātman, आत्मन्), and atoms of solid (pṛthvī, पृथ्वी), liquid (āpaḥ, आपः), radiant energy (tejas, तेजस्), gas (vāyu वायु), and mind (mana, मन).

According to physics ,solid, liquid, gas and plasma are mere different states of matter as also energy may mutually transform into each other. Therefore, they represent a single measurable fundamental physical quantity, ‘mass’ or padārtha (पदार्थ). The above mentioned states are only the representations of this fundamental quantity or padārtha. Mass is the physical quantity which can dwell in space-time continuum (Vibhu, विभु) and therefore the measurement of length vector and time become the second and third fundamental physical quantities, respectively. In physics, mass, length vector and time are the three fundamental physical quantities. Some more discussion is offered in the Vaiśeṣikaśāstra on the subject.

Samavāya (Concomitance):

There is the following comment given about concomitance in Nyāyakandalī (न्यायकन्दली):

अयुतसिद्धयोगाश्रयाश्रयिभावः समवाय इति |

एषस्समवायः तन्निष्ठगुणकर्मणां चाधाराधेयभावमूलकं सम्बन्धं स्फुटीकरोति |

वैशेषिकमतानुसारेण सामान्यतया द्वौ सम्बन्धौ

परिज्ञायेते, तौ च संयोगः समवायश्चेति |

Samavāya is a physical quality (guṇa, गुण) which exists non-conjunctively between the inheritor and the inherited. This concomitance explains a relation between the fundamental physical quantities and their properties, as well as the motion pertaining to them.

According to Vaiśeṣika in general, there are two relations: one is conjunctive (saṁyoga, संयोग) and the other is concomitance (samavāya). Temporary and momentary relation between any two substances is known as conjunctive relation. Boat and river relation is an example of conjunctive relation. Similarly, the relative relation between the fundamental physical quantity and its corresponding property is an example of concomitance. The physical quantity and its property, such as the relation between sugar and sweetness can be termed as concomitance, this relation is non-conjunctive and eternal which can never be separated, similarly, the sweetness can never be separated from the sugar. Concomitance is altogether different from the conjunctive relation. It is the relation of:

  • parts with the whole
  • properties and the motions with the fundamental physical quantities
  • elements with sets
  • specific with fundamental physical quantity.